- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Quantifying Robustness and Locality of Majorana Bound States in Interacting Systems

William Samuelson1,*, Juan Daniel Torres Luna2, Sebastian Miles2, A. Mert Bozkurt2, Martin Leijnse1, Michael Wimmer2, and Viktor Svensson3,†

  • *Contact author: william.samuelson@ftf.lth.se
  • †Contact author: viktor.svensson@fys.uio.no
PDF

PRX Quantum 7, 020341 – Published 28 May, 2026

DOI: https://doi.org/10.1103/pw51-tnsz

Abstract

Protecting qubits from perturbations is a central challenge in quantum computing. Topological superconductors with separated Majorana bound states (MBSs) provide a strong form of protection that only depends on the locality of perturbations. While the link between MBS separation, robust degeneracy, and protected braiding is well understood in noninteracting systems, recent experimental progress in short quantum-dot-based Kitaev chains highlights the need to establish these connections rigorously for interacting systems. We do this by defining MBSs from many-body ground states and show how their locality constrains their coupling to an environment. This, in turn, quantifies the protection of the energy degeneracy and the feasibility of non-abelian braiding.

View figure in article

Physics Subject Headings (PhySH)

Corrections

1 June, 2026

Correction: The affiliation indicator for the sixth author was incorrect and has been fixed.

Popular Summary

A central challenge in quantum computing is to mitigate the effect of noise. One possible solution is a quantum computer based on exotic quasiparticles known as Majorana bound states (or just Majoranas). A pair of spatially separated Majoranas would make a stable qubit. However, in systems where electrons interact by their charges, the Majoranas can no longer be described separately. It is then much harder to understand where they are located. Recent experiments in strongly interacting quantum-dot chains show clear signs of Majoranas but require theoretical insight. In our study, we develop the necessary framework, showing that separated Majoranas provide stable qubits, even when interactions cannot be neglected.

Instead of focusing on particles, which is common in condensed matter physics, we focus on quantum states. Such states are not localized anywhere; they describe the whole system. Nevertheless, we show that the story about separated Majoranas and stable qubits can be told solely from the states, without any reference to a particle.

Our framework provides a way to explore strongly interacting systems, such as the quantum-dot chains mentioned earlier. In the future, the framework could be extended to describe a broader range of systems. So far, we have focused on situations with two relevant quantum states, but an exciting next step would be to generalize the theory to handle several states.

Article Text

References (81)

  1. A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys.-Usp. 44, 131 (2001).
  2. F. Wilczek, Majorana returns, Nat. Phys. 5, 614 (2009).
  3. M. Leijnse and K. Flensberg, Introduction to topological superconductivity and Majorana fermions, Semicond. Sci. Technol. 27, 124003 (2012).
  4. J. Alicea, New directions in the pursuit of Majorana fermions in solid state systems, Rep. Prog. Phys. 75, 076501 (2012).
  5. R. Aguado, Majorana quasiparticles in condensed matter, Riv. Nuovo Cimento 40, 523 (2017).
  6. C. W. J. Beenakker, Search for Majorana fermions in superconductors, Ann. Rev. Condens. Matter Phys. 4, 113 (2013).
  7. R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures, Phys. Rev. Lett. 105, 077001 (2010).
  8. Y. Oreg, G. Refael, and F. von Oppen, Helical liquids and Majorana bound states in quantum wires, Phys. Rev. Lett. 105, 177002 (2010).
  9. S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A. Yazdani, Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor, Phys. Rev. B 88, 020407(R) (2013).
  10. M. Hell, M. Leijnse, and K. Flensberg, Two-dimensional platform for networks of Majorana bound states, Phys. Rev. Lett. 118, 107701 (2017).
  11. F. Pientka, A. Keselman, E. Berg, A. Yacoby, A. Stern, and B. I. Halperin, Topological superconductivity in a planar josephson junction, Phys. Rev. X 7, 021032 (2017).
  12. J. D. Sau and S. D. Sarma, Realizing a robust practical Majorana chain in a quantum-dot-superconductor linear array, Nat. Commun. 3, 964 (2012).
  13. I. C. Fulga, A. Haim, A. R. Akhmerov, and Y. Oreg, Adaptive tuning of Majorana fermions in a quantum dot chain, New J. Phys. 15, 045020 (2013).
  14. M. Leijnse and K. Flensberg, Parity qubits and poor man’s Majorana bound states in double quantum dots, Phys. Rev. B 86, 134528 (2012).
  15. C.-X. Liu, G. Wang, T. Dvir, and M. Wimmer, Tunable superconducting coupling of quantum dots via Andreev bound states in semiconductor-superconductor nanowires, Phys. Rev. Lett. 129, 267701 (2022).
  16. J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Non-abelian statistics and topological quantum information processing in 1D wire networks, Nat. Phys. 7, 412 (2011).
  17. D. Aasen, M. Hell, R. V. Mishmash, A. Higginbotham, J. Danon, M. Leijnse, T. S. Jespersen, J. A. Folk, C. M. Marcus, K. Flensberg, and J. Alicea, Milestones toward Majorana-based quantum computing, Phys. Rev. X 6, 031016 (2016).
  18. A. Tsintzis, R. S. Souto, K. Flensberg, J. Danon, and M. Leijnse, Majorana qubits and non-abelian physics in quantum dot–based minimal Kitaev chains, PRX Quantum 5, 010323 (2024).
  19. T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, C. M. Marcus, and M. H. Freedman, Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes, Phys. Rev. B 95, 235305 (2017).
  20. L. Fidkowski and A. Kitaev, Effects of interactions on the topological classification of free fermion systems, Phys. Rev. B 81, 134509 (2010).
  21. L. Fidkowski and A. Kitaev, Topological phases of fermions in one dimension, Phys. Rev. B 83, 075103 (2011).
  22. E. M. Stoudenmire, J. Alicea, O. A. Starykh, and M. P. A. Fisher, Interaction effects in topological superconducting wires supporting Majorana fermions, Phys. Rev. B 84, 014503 (2011).
  23. F. Hassler and D. Schuricht, Strongly interacting Majorana modes in an array of Josephson junctions, New J. Phys. 14, 125018 (2012).
  24. N. M. Gergs, L. Fritz, and D. Schuricht, Topological order in the Kitaev/Majorana chain in the presence of disorder and interactions, Phys. Rev. B 93, 075129 (2016).
  25. S. V. Aksenov, A. O. Zlotnikov, and M. S. Shustin, Strong Coulomb interactions in the problem of Majorana modes in a wire of the nontrivial topological class BDI, Phys. Rev. B 101, 125431 (2020).
  26. N. Chepiga and N. Laflorencie, Topological and quantum critical properties of the interacting Majorana chain model, SciPost Phys. 14, 152 (2023).
  27. H. Katsura, D. Schuricht, and M. Takahashi, Exact ground states and topological order in interacting Kitaev/Majorana chains, Phys. Rev. B 92, 115137 (2015).
  28. J.-J. Miao, H.-K. Jin, F.-C. Zhang, and Y. Zhou, Exact solution for the interacting Kitaev chain at the symmetric point, Phys. Rev. Lett. 118, 267701 (2017).
  29. G. Goldstein and C. Chamon, Exact zero modes in closed systems of interacting fermions, Phys. Rev. B 86, 115122 (2012).
  30. A. R. Wright and M. Veldhorst, Localized many-particle Majorana modes with vanishing time-reversal symmetry breaking in double quantum dots, Phys. Rev. Lett. 111, 096801 (2013).
  31. T. E. O’Brien, A. R. Wright, and M. Veldhorst, Many-particle Majorana bound states: Derivation and signatures in superconducting double quantum dots, Phys. Status Solidi B 252, 1731 (2015).
  32. T. E. O’Brien and A. R. Wright, A many-body interpretation of Majorana bound states, and conditions for their localisation, arXiv:1508.06638 [cond-mat].
  33. G. Kells, Many-body Majorana operators and the equivalence of parity sectors, Phys. Rev. B 92, 081401(R) (2015).
  34. G. Kells, Multiparticle content of Majorana zero modes in the interacting p-wave wire, Phys. Rev. B 92, 155434 (2015).
  35. M. McGinley, J. Knolle, and A. Nunnenkamp, Robustness of Majorana edge modes and topological order: Exact results for the symmetric interacting Kitaev chain with disorder, Phys. Rev. B 96, 241113(R) (2017).
  36. G. Kells, N. Moran, and D. Meidan, Localization enhanced and degraded topological order in interacting p-wave wires, Phys. Rev. B 97, 085425 (2018).
  37. A. M. Bozkurt, S. Miles, S. L. D. ten Haaf, C.-X. Liu, F. Hassler, and M. Wimmer, Interaction-induced strong zero modes in short quantum dot chains with time-reversal symmetry, SciPost Phys. 18, 206 (2025).
  38. V. Svensson and M. Leijnse, Quantum dot based Kitaev chains: Majorana quality measures and scaling with increasing chain length, Phys. Rev. B 110, 155436 (2024).
  39. T. Dvir, G. Wang, N. van Loo, C.-X. Liu, G. P. Mazur, A. Bordin, S. L. D. ten Haaf, J.-Y. Wang, D. van Driel, F. Zatelli, X. Li, F. K. Malinowski, S. Gazibegovic, G. Badawy, E. P. A. M. Bakkers, M. Wimmer, and L. P. Kouwenhoven, Realization of a minimal Kitaev chain in coupled quantum dots, Nature 614, 445 (2023).
  40. S. L. D. ten Haaf, Q. Wang, A. M. Bozkurt, C.-X. Liu, I. Kulesh, P. Kim, D. Xiao, C. Thomas, M. J. Manfra, T. Dvir, M. Wimmer, and S. Goswami, A two-site Kitaev chain in a two-dimensional electron gas, Nature 630, 329 (2024).
  41. F. Zatelli, D. van Driel, D. Xu, G. Wang, C.-X. Liu, A. Bordin, B. Roovers, G. P. Mazur, N. van Loo, J. C. Wolff, A. M. Bozkurt, G. Badawy, S. Gazibegovic, E. P. A. M. Bakkers, M. Wimmer, L. P. Kouwenhoven, and T. Dvir, Robust poor man’s Majorana zero modes using Yu-Shiba-Rusinov states, Nat. Commun. 15, 7933 (2024).
  42. A. Bordin, C.-X. Liu, T. Dvir, F. Zatelli, S. L. D. ten Haaf, D. van Driel, G. Wang, N. van Loo, Y. Zhang, J. C. Wolff, T. Van Caekenberghe, G. Badawy, S. Gazibegovic, E. P. A. M. Bakkers, M. Wimmer, L. P. Kouwenhoven, and G. P. Mazur, Enhanced Majorana stability in a three-site Kitaev chain, Nat. Nanotechnol. 20, 726 (2025).
  43. S. L. D. ten Haaf, Y. Zhang, Q. Wang, A. Bordin, C.-X. Liu, I. Kulesh, V. P. M. Sietses, C. G. Prosko, D. Xiao, C. Thomas, M. J. Manfra, M. Wimmer, and S. Goswami, Observation of edge and bulk states in a three-site Kitaev chain, Nature 641, 890 (2025).
  44. A. Bordin, F. J. Bennebroek Evertsz’, B. Roovers et al., Probing Majorana localization of a phase-controlled three-site Kitaev chain with an additional quantum dot, Nat. Commun. 17, 2313 (2026).
  45. J. Alicea and P. Fendley, Topological phases with parafermions: Theory and blueprints, Ann. Rev. Condens. Matter Phys. 7, 119 (2016).
  46. K. Wada, T. Sugimoto, and T. Tohyama, Coexistence of strong and weak Majorana zero modes in an anisotropic XY spin chain with second-neighbor interactions, Phys. Rev. B 104, 075119 (2021).
  47. P. Fendley, Strong zero modes and eigenstate phase transitions in the XYZ/interacting Majorana chain, J. Phys. A: Math. Theor. 49, 30LT01 (2016).
  48. J. Behrends, J. H. Bardarson, and B. Béri, Tenfold way and many-body zero modes in the Sachdev-Ye-Kitaev model, Phys. Rev. B 99, 195123 (2019).
  49. S. Szalay, Z. Zimborás, M. Máté, G. Barcza, C. Schilling, and Ö. Legeza, Fermionic systems for quantum information people, J. Phys. A: Math. Theor. 54, 393001 (2021).
  50. To simplify notation in the main text, we do not write explicitly on which space operators act. The convention we use is that it acts on the smallest space possible allowed in the context. When two operators in different spaces are multiplied or added, they are implicitly extended to a joint space by a fermionic tensor product.
  51. S. Bravyi, D. P. DiVincenzo, and D. Loss, Schrieffer–Wolff transformation for quantum many-body systems, Ann. Phys. 326, 2793 (2011).
  52. A. Tsintzis, R. S. Souto, and M. Leijnse, Creating and detecting poor man’s Majorana bound states in interacting quantum dots, Phys. Rev. B 106, L201404 (2022).
  53. R. S. Souto, A. Tsintzis, M. Leijnse, and J. Danon, Probing Majorana localization in minimal Kitaev chains through a quantum dot, Phys. Rev. Res. 5, 043182 (2023).
  54. W. Samuelson, V. Svensson, and M. Leijnse, Minimal quantum dot based Kitaev chain with only local superconducting proximity effect, Phys. Rev. B 109, 035415 (2024).
  55. J. D. Torres Luna, A. M. Bozkurt, M. Wimmer, and C.-X. Liu, Flux-tunable Kitaev chain in a quantum dot array, SciPost Phys. Core 7, 065 (2024).
  56. N. Sedlmayr and C. Bena, Visualising Majorana bound states in 1D and 2D using the generalized Majorana polarization, Phys. Rev. B 92, 115115 (2015).
  57. N. Sedlmayr, J. M. Aguiar-Hualde, and C. Bena, Majorana bound states in open quasi-1D and 2D systems with transverse Rashba coupling, Phys. Rev. B 93, 155425 (2016).
  58. C. Bena, Testing the formation of Majorana states using Majorana polarization, C. R. Phys. 18, 349 (2017).
  59. E. Prada, R. Aguado, and P. San-Jose, Measuring Majorana nonlocality and spin structure with a quantum dot, Phys. Rev. B 96, 085418 (2017).
  60. D. J. Clarke, Experimentally accessible topological quality factor for wires with zero energy modes, Phys. Rev. B 96, 201109(R) (2017).
  61. M. T. Deng, S. Vaitiekėnas, E. B. Hansen, J. Danon, M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, and C. M. Marcus, Majorana bound state in a coupled quantum-dot hybrid-nanowire system, Science 354, 1557 (2016).
  62. C.-X. Liu, A. M. Bozkurt, F. Zatelli, S. L. D. ten Haaf, T. Dvir, and M. Wimmer, Enhancing the excitation gap of a quantum-dot-based Kitaev chain, Commun. Phys. 7, 235 (2024).
  63. A. E. G. Mikkelsen, P. Kotetes, P. Krogstrup, and K. Flensberg, Hybridization at superconductor-semiconductor interfaces, Phys. Rev. X 8, 031040 (2018).
  64. A. E. Antipov, A. Bargerbos, G. W. Winkler, B. Bauer, E. Rossi, and R. M. Lutchyn, Effects of gate-induced electric fields on semiconductor Majorana nanowires, Phys. Rev. X 8, 031041 (2018).
  65. Q. Wang, S. L. D. ten Haaf, I. Kulesh, D. Xiao, C. Thomas, M. J. Manfra, and S. Goswami, Triplet correlations in Cooper pair splitters realized in a two-dimensional electron gas, Nat. Commun. 14, 4876 (2023).
  66. A. Bordin, G. Wang, C.-X. Liu, S. L. D. ten Haaf, N. van Loo, G. P. Mazur, D. Xu, D. van Driel, F. Zatelli, S. Gazibegovic, G. Badawy, E. P. A. M. Bakkers, M. Wimmer, L. P. Kouwenhoven, and T. Dvir, Tunable crossed andreev reflection and elastic cotunneling in hybrid nanowires, Phys. Rev. X 13, 031031 (2023).
  67. J. Danon, A. B. Hellenes, E. B. Hansen, L. Casparis, A. P. Higginbotham, and K. Flensberg, Nonlocal conductance spectroscopy of Andreev bound states: Symmetry relations and BCS charges, Phys. Rev. Lett. 124, 036801 (2020).
  68. T. Ö. Rosdahl, A. Vuik, M. Kjaergaard, and A. R. Akhmerov, Andreev rectifier: A nonlocal conductance signature of topological phase transitions, Phys. Rev. B 97, 045421 (2018).
  69. J. D. Sau, D. J. Clarke, and S. Tewari, Controlling non-abelian statistics of Majorana fermions in semiconductor nanowires, Phys. Rev. B 84, 094505 (2011).
  70. C. W. J. Beenakker, Search for non-abelian Majorana braiding statistics in superconductors, SciPost Phys. Lect. Notes 15 (2020).
  71. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).
  72. M. Nitsch, V. Svensson, W. Samuelson, K. Nestmann, J. Danon, K. Flensberg, R. S. Souto, and M. Leijnse, Adiabatic nonabelian braiding of imperfect Majoranas, arXiv:2507.11039 [cond-mat].
  73. S. Miles, F. Zatelli, A. M. Bozkurt, M. Wimmer, and C.-X. Liu, Braiding Majoranas in a linear quantum dot-superconductor array: Mitigating the errors from Coulomb repulsion and residual tunneling, Phys. Rev. B 113, 085302 (2026).
  74. G. Ortiz, J. Dukelsky, E. Cobanera, C. Esebbag, and C. Beenakker, Many-body characterization of particle-conserving topological superfluids, Phys. Rev. Lett. 113, 267002 (2014).
  75. G. Ortiz and E. Cobanera, What is a particle-conserving topological superfluid? The fate of Majorana modes beyond mean-field theory, Ann. Phys. 372, 357 (2016).
  76. W. Samuelson, J. D. Torres Luna, S. Miles, A. M. Bozkurt, M. Leijnse, M. Wimmer, and V. Svensson, Quantifying robustness and locality of Majorana bound states in interacting systems, Zenodo, 2025, https://doi.org/10.5281/zenodo.17422940.
  77. A. E. Rastegin, Relations for certain symmetric norms and anti-norms before and after partial trace, J. Stat. Phys. 148, 1040 (2012).
  78. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, UK, 2010).
  79. P. C. Rico, New partial trace inequalities and distillability of Werner states, Lett. Math. Phys. 115, 47 (2025).
  80. H. Bruus, K. Flensberg, H. Bruus, and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics: An Introduction, Oxford Graduate Texts (Oxford University Press, 2004).
  81. R. A. Dourado, J. Danon, M. Leijnse, and R. S. Souto, Measuring coherence factors of states in superconductors through local current, Phys. Rev. B 112, 174515 (2025).

Outline

Information

PRX Quantum 7, 020341– Published 28 May, 2026
  • Received 31 October 2025
  • Revised 2 March 2026
  • Accepted 3 April 2026
  • Corrected 1 June 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation